Two-weight norm inequalities for the Cesàro means of Hermite expansions

Abstract

Abstract. An accurate estimate is obtained of the Cesàro kernel for Hermite expansions. This is used to prove two-weight norm inequalities for Cesàro means of Hermite polynomial series and for the supremum of these means. These extend known norm inequalities, even in the single power weight and "unweighted" cases. An almost everywhere convergence result is obtained as a corollary. It is also shown that the conditions used to prove norm boundedness of the means and most of the conditions used to prove the boundedness of the Cesàro supremum of the means are necessary.

The Bochner–Riesz means for Fourier–Bessel expansions: Norm inequalities for the maximal operator and almost everywhere convergence

“…They also obtain similar estimates for Cesàro means of Hermite polynomial series and for the supremum of those means in [15]. An almost everywhere convergence result is obtained as a corollary in [14] and [15]. The result about Laguerre polynomials is an extension of a previous result in [18].…”

“…For p = 1, they prove a weak type result. They also obtain similar estimates for Cesàro means of Hermite polynomial series and for the supremum of those means in [15]. An almost everywhere convergence result is obtained as a corollary in [14] and [15].…”

“…it is clear, from (15), that K δ R (x, y) = (xy) −(ν+1/2) K δ R (x, y). Then, it is verified that the inequality…”

Two-weight norm inequalities for the Cesàro means of generalized Hermite expansions

“…These kind of questions have been widely studied for the classical Hermite system [1,5,8,7], but not for the generalized Hermite system. This is the aim of this paper.…”

Norm inequalities for the conjugate operator in two-weighted Lebesgue spaces

“…In succession of classical weighted-norm inequalities, starting from Muckenhoupt's result in 1975 [6], there have been extensive studies on two-weighted norm inequalities (for textbooks [7][8][9][10] and for related topics [11][12][13][14][15][16][17]). In [6], Muckenhoupt derives a necessary and sufficient condition on two-weighted norm inequalities for the Poisson integral operator.…”